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Locally constrained graph homomorphisms

Locally constrained graph homomorphisms. Jan Kratochvíl Charles University, Prague. Outline of the talk. Graph h omomorphism

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Locally constrained graph homomorphisms

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  1. Locally constrained graph homomorphisms Jan Kratochvíl Charles University, Prague

  2. Outline of the talk • Graph homomorphism • Local constraints - graph covers partial covers – frequency assignment role assignments • Complexity results and questions

  3. 1. Graph homomorphism Edge preserving vertex mapping between graphs G and H f : V(G)  V(H) s.t. uv  E(G)  f(u)f(v) E(H)

  4. v f(v) f u f(u) H G

  5. H-COLORING Input: A graph G. Question:  homomorphism G H? Thm (Hell, Nešetřil): H-COLORING is polynomial for H bipartite and NP-complete otherwise.

  6. 2. Local constraints For every u V(G), f(NG(u))  NH(f(u))

  7. f f(u) u H G

  8. Definition: A homomorphism f : G  H is called bijective locally injective if for every u  V(G) surjective the restricted mapping f : NG(u))  NH(f(u)) bijective is injective . surjective

  9. 2. Locally constrained homomorphisms • loc. bijective = graph covers • loc. injective = partial covers = generalized frequency assignment • loc. surjective = role assignment • computational complexity

  10. 2.1 Locally bijective homomorphisms = graph covers • Topological graph theory, construction of highly symmetric graphs (Biggs, Conway) • Degree preserving

  11. 2.1 Locally bijective homomorphisms = graph covers • Topological graph theory, construction of highly symmetric graphs (Biggs, Conway) • Degree preserving • Local computation (Angluin, Courcelle)

  12. 2.1 Locally bijective homomorphisms = graph covers • Topological graph theory, construction of highly symmetric graphs (Biggs, Conway) • Degree preserving • Local computation (Angluin, Courcelle) • Degree partition preserving

  13. Degree partition – the coarsest partition V(G) = V1  V2  …  Vt s.t. there exist numbers rij s.t. | N(v)  Vj | = rij for every v  Vi .

  14. 2.1 Locally bijective homomorphisms = graph covers • Topological graph theory, construction of highly symmetric graphs (Biggs, Conway) • Degree preserving • Local computation (Angluin, Courcelle) • Degree partition preserving • Finite planar covers

  15. Conjecture (Negami): A graph has a finite planar cover if and only if it is projective planar.

  16. Conjecture (Negami): A graph has a finite planar cover if and only if it is projective planar. Attempts to prove via forbidden minors for projective planar graphs (Negami, Fellows, Archdeacon, Hliněný)

  17. Conjecture (Negami): A graph has a finite planar cover if and only if it is projective planar. Attempts to prove via forbidden minors for projective planar graphs (Negami, Fellows, Archdeacon, Hliněný) True if K1,2,2,2 does not have a finite planar cover.

  18. 2.2 Locally injective homomorphisms = partial covers Observation: A graph G allows a locally injective homomorphism into a graph H iff G is an induced subgraph of a graph G’ which covers H fully.

  19. 2.2 Locally injective homomorphisms = generalized frequency assignment

  20. L(2,1)-labelings of graphs (Roberts; Griggs, Yeh; Georges, Mauro; Sakai; Král, Škrekovski)

  21. L(2,1)-labelings of graphs f: V(G) {0,1,2,…,k} uv  E(G)  |f(u) – f(v)|  2 dG(u,v) = 2  f(u)  f(v)

  22. L(2,1)-labelings of graphs f: V(G) {0,1,2,…,k} uv  E(G)  |f(u) – f(v)|  2 dG(u,v) = 2  f(u)  f(v) |f(u) – f(v)|  1

  23. L(2,1)-labelings of graphs f: V(G) {0,1,2,…,k} uv  E(G)  |f(u) – f(v)|  2 dG(u,v) = 2  f(u)  f(v) |f(u) – f(v)|  1 L(2,1)(G) = min such k

  24. L(2,1)-labelings of graphs • NP-complete for every fixed k  4 (Fiala, Kloks, JK) • Polynomial for graphs of bounded tree-width (when k fixed)

  25. L(2,1)-labelings of graphs • NP-complete for every fixed k  4 (Fiala, Kloks, JK) • Polynomial for graphs of bounded tree-width (when k fixed) • Polynomial for trees when k part of input (Chang, Kuo) • Open for graphs of bounded tree-width (when k part of input)

  26. H(2,1)-labelings of graphs (Fiala, JK 2001)

  27. H(2,1)-labelings of graphs (Fiala, JK 2001) Ck(2,1)-labelings have been considered by Leese et al.

  28. H(2,1)-labelings of graphs f: V(G) V(H) uv  E(G)  dH (f(u), f(v))  2 dG(u,v) = 2  f(u)  f(v)

  29. H(2,1)-labelings of graphs f: V(G) V(H) uv  E(G)  dH (f(u), f(v))  2  f(u)f(v)  E(H) dG(u,v) = 2  f(u)  f(v)

  30. H(2,1)-labelings of graphs f: V(G) V(H) uv  E(G)  f(u)f(v)  E(-H) dG(u,v) = 2  f(u)  f(v)

  31. H(2,1)-labelings of graphs f: V(G) V(H) uv  E(G)  f(u)f(v)  E(-H) homomorphism from G to -H dG(u,v) = 2  f(u)  f(v) locally injective

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